#72220796

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×C₄⋊C₄ ⋊ C₂×C₄
Sylow	C₅² ⋊ C₄×#102

Subgroups

	subgroup	G/H
center	C₂²	#11717375
commutator	C₄×C₅²	C₂×C₄²
Frattini	C₂×C₄	C₂×D₅⋊AGL(1,𝔽₅)
Fitting	C₅²×C₄⋊C₄	C₂×C₄
socle	C₂²×C₅²	C₄×D₄
prime radical	C₂×#87877	C₂²
cosocle	C₂×C₄×C₅⋊D₅	C₂³
hypercenter	C₄⋊C₄	D₅⋊AGL(1,𝔽₅)
hypocenter	C₅²	C₄×#102
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

986 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 986 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^7,5^2|a^2=d,b^2=dfg,c^2=e,d^2=g,f^2=g,b^a=bf,d^b=dg,f^b=fg,h^a=h^4,h^c=h^3,h^e=h^4,i^a=h^2i,i^c=i^3,i^e=i^4

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220815, #72253303, #72253330, …
maximal normal subgroup of	#72220815, #72253303, #72253330, …
maximal quotient of	#72220797

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 5, 5]);;
gap> SetPower(coll, 1, F.4);;
gap> SetPower(coll, 2, F.4*F.6*F.7);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> SetConjugate(coll, 8, 3, F.8^3);;
gap> SetConjugate(coll, 8, 5, F.8^4);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 3, F.9^3);;
gap> SetConjugate(coll, 9, 5, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);